# Z-Scores

Z Scores were a concept I had trouble with in University. It’s actually not as difficult as they’re made out to be. I’ll spare you the complicated introduction (as I’m sure you got one from both your textbook and your Professor), but remember that z-scores show you the distance between your score and the mean, in standard deviations.

So a z-score of 1 is one standard deviation (approximately 84%) above the entire population, and 34% above the mean. Typically you’ll be asked to do a few things:

• What percent above the mean is a particular z-score
• What percent below the mean is a particular z-score
• What percent is between two scores
• How do you convert a raw score into a z-score
• How do you convert a z-score into a raw score

So, let’s get too it. Remember that you’ll need a z-table (usually provided by your Professor or available in you textbook) for these exercises.

## Raw Score into Z-Score

The formula for converting a raw score into a z-score is Z = (M – X) / SD, or Z Score = (Mean – Value) / Standard Deviation.

So, if you have a score of 80, and the mean is 75, with a Standard Deviation of 5, your equation will be:

(80 – 75) / 5 = 1. Therefore your Z score is 1.0

If you instead scored 73, it would be (73 – 75) / 5 = -0.4.

## Percent Below a Score

You’ll be given a z-score like 0.66, and you’ll need to find out what percent of scores are above it. Simply go to your z-table, and find 0.66. Some tables list all the values sequentially (0.5, 0.51, 0.52) while others use a table like Wikipedia’s.

If your table includes both “% mean to z” and “% in tail” (like my textbook), just look a the “% mean to z.” If your table uses decimals (like 0.7454), multiply them by 100 to get the correct value.

When I look up 0.66 in my textbook’s z-table, I see 24.54%. When I look up the same value in Wikipedia’s table, I see 74.54%. What gives? The % mean to z is only half of the equation. In order to return a correct z-score, you take your 24.54% and add 50 to it, because it’s a positive z-score.

If you have a negative z-score, like -0.85, you take the z-score you’re given (30.23%) and you subtract 50 from it, which gives you -19.77 (ignore the negative.), or 19.77%.

## Percent Above a Score

To find the percent above a score, you perform the same calculation for percent below, but you subtract from 100. For instance, 19.77% is percent below, so when you subtract that value from 100, you get 80.23% above.

## Percent Between Scores

To calculate percent between scores, you simply take the difference between two scores and subtract them. For instance, if you want to know the difference between 0.6 and 0.7:

% to mean of 0.6 is 22.57 and of 0.7 is 25.80.

Because both numbers are above the mean, we add 50 to each, giving us percentages of 72.57 and 75.80.

Subtracting 72.57 from 75.80 gives a grand total of 3.23% between the two values.

## Z-Score Back to Raw Score

The formula for converting a z-score back to a raw score is R = Z*SD + M. So if your z score is 0.8, the mean is 75 and the Standard Deviation is 5, your equation looks like:

Raw Score = 0.8*5 + 75

Raw Score = 4 + 75

Raw Score = 79

Cite this article as: MacDonald, D.K., (2015), "Z-Scores," retrieved on September 19, 2019 from http://dustinkmacdonald.com/z-scores/.

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